Question

An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs).

(only need c,e,and f answered)

x	17	34	49	28	50	25
y	3	3	6	5	9	3

Complete parts (a) through (e), given Σx = 203, Σy = 29, Σx² = 7755, Σy² = 169, Σxy = 1112, and r ≈ 0.818.

(a) Draw a scatter diagram displaying the data.

(b) Verify the given sums Σx, Σy, Σx², Σy², Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.)

(c) Find x, and y. Then find the equation of the least-squares line y hat = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)

x	=
y	=
y hat	=	+ x

(d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.

(e) Find the value of the coefficient of determination r². What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r² to three decimal places. Round your answers for the percentages to one decimal place.)

r2 =
explained	%
unexplained	%

(f) For a neighborhood with x = 40 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.)
   ____ hundred jobs

Answer 1

a.

b.

X Values
∑ = 203
Mean = 33.833
∑(X - M_x)² = SS_x = 886.833

Y Values
∑ = 29
Mean = 4.833
∑(Y - M_y)² = SS_y = 28.833

X and Y Combined
N = 6
∑(X - M_x)(Y - M_y) = 130.833

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 130.833 / √((886.833)(28.833)) = 0.818

c.

Sum of X = 203
Sum of Y = 29
Mean X = 33.8333
Mean Y = 4.8333
Sum of squares (SS_X) = 886.8333
Sum of products (SP) = 130.8333

Regression Equation = ŷ = bX + a

b = SP/SS_X = 130.83/886.83 = 0.148

a = M_Y - bM_X = 4.83 - (0.15*33.83) = -0.158

ŷ = 0.148X - 0.158

d.

e. r^2=0.818^2=0.669

So Explained variation is 66.9%

Unexplained variation is 33.1%

f. For x=40,

ŷ = (0.148*40) - 0.158=5.76